On the complexity of deciding fair termination of probabilistic concurrent finite-state programs

Abstract

In this paper, we consider the fair termination problem for probabilistic concurrent finite-state programs. We analyse the complexity of deciding, given a system of probabilistic concurrent finite-state programs, whether the system will terminate (with probability 1) under five different fairness assumptions. It turns out that these problems are either complete for Ptime, or the second or third level of the alternating logspace hierarchy, even when the number of programs is fixed. However, when a more succinct representation of the inputs is allowed, the problems become either Exptime- or Pspace-complete. Let NL k (AL 2 k , AL 3 k , respectively) denote the class of languages that can be accepted by a k log n space bounded NTM (2-alternation bounded ATM, 3-alternation bounded ATM, respectively) with a binary tape alphabet. It has been conjectured that there are problems in NL k that require Ω( n k ) deterministic time. Now these problems are in Ptime for every fixed value of k; but if we accept the conjecture, then the order of the polynomial, in each case, grows linearly with the value of k. For succinctly represented instances of k concurrent n-state programs, we show that the problems require Ω(n (k−6) 96 ) deterministic time or are hard for AL 2 k−2 or AL 3 k−3 . Furthermore, the reductions are such that if we accept the conjecture, then each of the problems require Ω(n (k−2) 8 ) deterministic time. In addition, corresponding upper bounds of O( n d k ) deterministic time and d k log n 2- (3-)alternation bounded ATM space (for some positive constant d) are shown; and lower bounds with respect to space are discussed.